Strategic Bidding in a Pool-based Electricity Market Under
Load Forecast Uncertainty
Shahnam Gorgizadeh
Asghar Akbari Foroud
Meisam Amirahmadi
Electrical & Computer
Engineering Faculty, Semnan
University, Iran
[email protected]
Electrical & Computer
Engineering Faculty, Semnan
University, Iran
[email protected]
Electrical & Computer
Engineering Faculty, Semnan
University, Iran
[email protected]
Abstract: This paper proposes a method for participants in an imperfect electricity market to bid
with taking load forecast uncertainty and demand response into consideration. For analyzing the
effect of load uncertainty on participants’ payoffs, the equation between load and payoffs in
different situations of demand response is obtained, and it is shown that if the load is stochastic
how much it can affect the mathematical expectancy of an agent’s payoff in an unconstrained
power grid. The method used for bidding strategy is based on the fact that the market is like a
repeated game. Therefore, participants should play cooperatively with each other. Consequently,
they must maximize other participants’ payoffs in addition to theirs. In this method Genetic
Algorithm and Fuzzy sets are used for maximizing payoffs, and for considering load uncertainty
in bidding strategy process, Mont-Carlo method has been combined with above method.
Key Words: Electricity market, Bidding strategy, Game theory, Genetic algorithm, Fuzzy sets,
Mont-Carlo simulation
Nomenclature
Nl
Set of all branches
Pg ,i
Generation of a generator
Nl(l)
Set of branches in independent loop l
p dg,i
Generation of a DG
Ndco
Set of all Discos
p IL,i
IL granted to a Disco
Ngco
Set of all Gencos
Pij
Power flow of line ij
Sn  m 
Set of buses for Disco m
Rd
Ng n
Set of generators for Genco n
Rg i
Profit of Genco i
Ndg
Set of all DGs
 (R i )
Profit’s membership function
NIL
Set of all ILs
C( Pg,i )
Cost function of a Genco
Nlp
Set of independent loops
C( Pdg,i )
Cost function of a DG
n
Set of all buses
C(PIL,i )
Cost function of Disco’s IL
n(i)
Set of buses connected to bus i
Pg,mini , Pg,max
i
Generator’s lower and upper limits
0
Pd,i
Total demand of a disco
min
Pdg,
i
DG’s lower and upper limits
max
Pij
Maximum flow limit on line ij
min
max
PIL,i
, PIL,i
IL’s lower and upper limits
x ij
Reactance of line ij
a dg,i , b dg,i
Generation cost coefficients of a DG
c
Disco’s retail energy price
a IL,i , b IL,i
IL cost coefficients of a Disco
k g ,i
Genco’s strategic parameter
a g,i , b g,i , cg,i
Cost coefficients of a generator
I.
Profit of Disco i
i
,
max
Pdg,
i
Introduction
In recent decades, the electricity supply industry throughout the world has been moved from centralized
and vertically integrated structure to an open market environment. The main objective of these markets is
to decrease the cost of electricity through competition [1]. There are various models for an electricity
market e.g. wholesale market. In this model no central organization is responsible for the provision of
electricity energy. Instead, Discos purchase the electrical energy consumed by their customers directly
from generating companies. These transactions which take place in a wholesale electricity market can
take the form of a pool or of a bilateral transaction [2]. In a bilateral transaction only buyers and sellers
are involved, and the price of each transaction is set independently by the parties involved. This kind of
transaction is appropriate for parties wanting to hedge against the price volatility in other markets. In [3]
bidding strategies in a bilateral market was examined. In a pool market, the producers and consumers
submit their bids, including a set of quantities with their prices, to market operator. Then the market
clearing prices (MCP) is announced by the market operator. Regardless of the bidding prices from
suppliers, all selected bidders are paid the MCP. This approach is adopted to encourage suppliers in a
competitive market to price energy close to their marginal costs [4]. In such a market, each agent tries to
establish a suitable bidding strategy to maximize its profit. Finding the best method to bid in such a
market is dependent on the type of competition. In a perfect competition, all participants are price-takers
that is no participant can influence the market price unilaterally. Theoretically, in perfectly competitive
market, suppliers should bid at, or very close to, their marginal production costs to maximize returns [4].
In such a competition, price forecasting is crucial for participants. In [5] and [6] Neural networks were
used to predict prices in England & Wales pool and in California market respectively. Fuzzy neural
network was used in [7] to forecast short-term price in an electricity market, and [8] provides a bidding
rule that allows a price taker producer to achieve, under price uncertainty, its optimal self-schedule.
However, electricity markets are more akin to oligopoly than perfect competitive environment. In an
imperfect competition, one company’s actions have significant influence on the profit of other companies.
Therefore, each company in establishing its bidding strategy for maximizing its profit must consider other
companies’ actions in the market. In such competition, game theory would be helpful. In [9], competition
among pool participants was modeled as a non-cooperative game with incomplete information, and in
[10] game theory was used by the independent system operator (ISO) to identify coalitions among
participants. In [11], a linear supply function model was presented to investigate strategic bidding
behavior and to illustrate some of the ways market power can be exercised. A more detailed review of
published work related to the generation of bidding strategies is given in [12].
It is widely recognized that markets will work better when demand response is available. Interrupting
loads (ILs) and distributed generations (DGs) are two major resources of demand response which Discos
can exploit to improve their market activity and increase their payoffs. In [13] a method for maximizing
payoff of a Disco which has some DGs was presented but the structure of wholesale market was not
regarded. In [14], a Disco for acquiring energy in a wholesale market uses DGs and ILs to minimize its
costs, but the impact of other Discos’ strategies is not considered in that paper. [15] presents a multiperiod energy acquisition model for a Disco trying to maximize its payoff by using DGs and ILs and
taking other Discos’ strategies into consideration.
In this paper a day-ahead pool market is considered in which Discos and Gencos trade energy in an
imperfect competition. The general structure of this market is similar to [16]. In this paper each Genco
tries to maximize its profit by establishing its generators supply curves, and each Disco pursues a similar
goal through using DGs and ILs. Therefore, Discos are not passive in the market and can influence the
market clearing price. Since power transaction in a power market occurs frequently, power market could
be considered as a repeated game. Therefore, a method for bidding in such market is presented.
Accordingly, each participant in addition to maximizing its profit must maximize other participants’
profits according to their ability to influence MCPs. Therefore, the strategic bidding problem is
formulated as a bilevel optimization problem. In this bilevel optimization problem, the upper level subproblem maximizes participants’ payoffs and the lower sub-problem solves the independent system
operator’s (ISO) market clearing problem. Genetic algorithm and Fuzzy sets is utilized for solving this
problem.
One of the common uncertainties in power systems is the uncertainty of load forecasting. This uncertainty
can affect companies’ estimation of their profits. Consequently it can influence companies’ bidding
strategy. In this paper the equation between load and Gencos’ and Discos’ profits is obtained, and it is
demonstrated that, in an unconstrained power network, how much load uncertainty can influence profit
of a risk neutral participant, whose mathematical expectation of his profit is important to him, and is
indifferent to the risk involved in his profit, and it will be obtained that if the load is considered as a
random variable with normal probability distribution, mathematical expectation of Gencos’ and Discos’
profits will increase and decrease respectively as the standard deviation in normal probability distribution
function increases. It also will be shown that if there is line flow capacity constraint in power network,
the relation between companies’ expected profits and standard deviation of uncertain load will completely
be dependent on companies’ location in power network, and some companies’ profits show high
sensitivity to standard deviation of load, in such situations load uncertainty in bidding strategy process
must be considered more carefully. Therefore, for considering load uncertainty in bidding strategy
process Mont-Carlo method is combined with the above mentioned method.
This paper is organized as follows. In sections ΙΙ and ΙΙΙ, Gencos’ and Discos’ profit model are formulated
and their strategies in the market are presented. Section ΙV describes the market clearing model. In section
V the bidding model is obtained. The relation between load and companies’ profits is formulated in VΙ,
the algorithm for solving the bidding model is presented in VΙΙ, and finally a numerical example surveys
the algorithm in section VΙΙΙ.
II.
Gencos’ profit model and strategies
It is assumed that each Genco has Ng n generators whose cost functions are according to the following:
C
g, i
2
 C(P )  a P  b P  c
g, i
g, i g, i
g, i g, i
g, i
(i  1, 2,
,
, Ng n )
(1)
Therefore, generators marginal cost will be:
MC  2a P  b
i
g,i g,i
g,i
(2)
A generator’s marginal cost is a linear function of power generated by generator ( Pg,i ), and is unit’s linear
bid curve under perfect competition.
The variation in bidding will be modeled as the variation of a single parameter k , multiplied by the
generators’ marginal cost. Therefore, a generator’s supply function could be modeled as:
bid(P )   (P )  k .(2a P  b )
g,i
g,i g,i
g,i
g,i g,i
g,i
min
(3)
max
Each Genco submits its supply curve, Pg,i and Pg ,i
to the ISO, and ISO clears the market using a
security constraint economic dispatch and determines each generator’s production amount and local
marginal prices (LMPs). According to the LMPs, each Genco’s payoff could be formulated as:
Ngn
Ngn
2
Rg n   LMPj . Pg,j   (a g,jPg,j  b jPg,j  cg,j )
j1
j1
, n  Ngco
(4)
Discos’ profit model and strategies
III.
The objective of a Disco is to maximize its profit by scheduling its DGs. A Disco does not bid its DGs
into the day-ahead market but schedules it’s DGs according to estimated LMPs, and also utilizes ILs to
restrain high LMPs.
It is assumed that the cost function of a DG is:
2
C(Pdg,i )  a dg,i . Pdg,i  bdg,i .pdg,i
, i  Ndg
(5)
And the cost curve of a customer to curtail its load is given as:
2
C(PIL,i )  a IL,i . PIL,i  b IL,i .p IL,i
, i  NIL
(6)
We assume that if the load of a Disco is interrupted, the Disco will be paid according to LMP and load
reduction, and the Disco returns all the compensation collected from the interruption of load to the
interrupted end customer and does not benefit from the IL compensation. Once the ISO cleared the
market, every Disco pays the ISO according to LMP and the energy purchased from the market. The cost
of purchased energy is:
 
C Pd,i  LMPi  Pd,i
, i  Sn  m 
(7)
Where Pd ,i is the energy purchased from the market by a Disco at bus i . Therefore, a Disco’s profit is the
difference between the revenue it collects from customers and the cost it pays to purchase energy from
market and produce energy by its DGs.
The profit of a Disco can be formulated as follows:
Rd m 

  


0
0
 c Pd,i  PIL,i  C Pdg ,i  LMPi Pd,i  PIL,i  Pdg,i 


i Sn(m)
, m  Ndco
(8)
IV.
Market clearing model
According to what mentioned before, each Genco submits its supply curve and each Disco submits its
demand, lower and upper ILs limits, and cost curves for ILs to ISO. Then ISO clears the market using a
security constraint economic dispatch. The objective in this level is to minimize generation costs and the
cost for compensating ILs subject to the bids and line constraints:
2
min(  (k g, i . a g,i . Pg,i
 k g, i .bg, i .Pg, i )   CIL (PIL , i ))
i  Ng
i  Nil
(9)
Subject to:
min
max
Pg,
i  Pg, i  Pg, i ,
i  Ng
(10)
ij  Nl
(11)
min
max
PIL,i
 PIL,i  PIL,i
, i  NIL
(12)
 Pd,i  PIL,i  Pdg,i   Pg,i  jn(i) Pij  0 , i  n
(13)
max
Pij
max
 Pij  Pij
,
0

P  x ij  0 , l  Nlp
(14)
ijNl(l) ij
Constraints (10)-(12) represent generation capacity constraint, transmission line constraint, and IL
capacity constraint respectively. Constraint (13) is the load balance constraint on individual buses, and
constraint (14) means that the summation of branch voltages in any independent loop must be equal to
zero, this constraint is regarded according to DC power flow. By solving this optimization problem, ISO
determines Pg, i , PIL , i and LMP on each bus which is the Lagrangian multiplier of constraint (13).
V.
Participants’ bidding model
In previous sections, we obtained each agent’s profit and strategy model, and as mentioned each agent
tries to maximize its profit. Since the competition is imperfect, each agent’s strategy will influence other
agents’ profits. By considering game theory, we will understand that if the competition is considered as a
non-repeated game, since there is no future for the game, each player follows its momentary profit and
plays non-cooperatively in the game. Therefore, the game will be balanced in a point which does not
necessarily have the best payoff for players [17]. Due to the fact that a day-ahead market occurs every
day, power market could not be categorized in such games, and could be considered as an infinitely
repeated game. In such game, it is better for each player to plays cooperatively with other players. On the
other hand, according to [18], market regulations should prevent any coalition except the grand coalition
and encourage cooperation in the power market; therefore, each player must maximize its profit, along
with other players’ profit. According to these reasons, each participant, for determining its bid, should
solve a bilevel optimization problem whose upper level sub-problem maximizes all participants’ profit,
and in lower level sub-problem market clearing model is considered. These can be formulated as follows:
 max Rd1
 max Rd
2


 max Rd

Ndco
 max Rg
1

 max Rg 2



 max Rg Ngco
(15)
Subject to:
min
max
Kg,
i  Kg, i  Kg, i
,i  1,
, Ng
(16)
min
max
Pdg,
i  Pdg, i  Pdg, i
, i  1,
, Ng
(17)
(9) - (14)
(18)
Each player solving this problem tries to find a combination of participants’ strategies to maximize (14),
and after finding this combination he will pick its strategy from it.
Since one of the common uncertainties in power systems is load uncertainty, and it can influence
participants’ bidding strategy, each participant should be aware of the load uncertainty’s influence on its
profit and arrange its bid according to this influence. Therefore, before proposing a method for solving
bidding strategy problem, we will establish the relation between each participant’s profit and load, and
then we will obtain load uncertainty’s impact on profit of a risk neutral participant, and finally a model
for solving bidding strategy problem with taking consideration of load uncertainty will be presented.
VI.
Modeling participants’ profit with load
For analyzing the relation between participants’ profit with load, we assume a simple system without any
line flow capacity constraint. In this system, there is a Genco which has a generator with a cost function
according to (1) and submits (3) as its bid, and also it is assumed that there is a Disco which has an IL
with a cost function according to (6), and therefore IL’s marginal cost is 2 a IL PIL  b IL . If assumed that
IL has a lower and upper limit (PILmin  PIL  PILMax ) , the profits of participants could be derived
according to Fig.1. Curve no.1 in Fig.1 is generator’s marginal cost curve and curve no.2 represents its
bidding curve, L is the amount of load, and L  PILMax is the amount of load after interrupting IL fully
and c is Disco’s retail price. If curve no.2 crosses load in a point between L and L  PILmin , load could
be considered as L or L  PILmin depending on the IL’s plan of action. It should be noticed that fixed
cost coefficient ( c ) is not considered in Genco’s profit in Fig.1 and must finally be subtracted from
Genco’s profit.
Generally for considering load uncertainty a normal probability distribution function is applied [19].
Since the load is a random variable with specific probability distribution, to find participants’ expected
profit is like to oscillate load curve in Fig.1 to left and right according to load’s randomness pattern and
averaging Genco’s and Disco’s profit. Therefore, the standard deviation in load’s probability distribution
could affect participant’s expected profit. For determining this impact, since there is IL in the system, the
problem should be analyzed in following states:
1. No load is interrupted
2. Part of IL is interrupting
3. IL is fully interrupted
4. There are multiple suppliers in system
(It is assumed that the lower limit of IL is zero)
λc
bIL+2aILPILMax
bIL
PILmin
K(2aL+b)
kb
b
L-PILMax
Profit of Genco
L
Power(MW)
Profit of Disco
Fig.1. Companies profits in a simple market
State 1- No load is interrupted
In this state, no load is curtailed with respect to Genco’s bid, the condition for being in this state is:
b kb
k(2aL  b)  b IL  L  IL
2a
(19)
And by considering Fig.1, Genco’s profit could be obtained as follows:
Rg  L(2kaL  bk)  (aL2  bL  c)  (2ka  a)L2  (kb  b)L  c
(20)
This equation shows that profit in this state is a quadratic function of load. Therefore, by taking random
sample of load, the expected profit of Genco will be more than the situation when load is considered its
2
mean value. For example, we consider a quadratic function f (x)   x   x  c , and we assume that x
has a normal probability distribution that x 0 and  denote its mean value and standard deviation
respectively. If we want to calculate expected value of f (x) by taking random samples of x , for every
two samples that have  difference with x 0 we will have:
2
2
f (x 0   )  f (x 0   )  (x 0   )   (x 0   )  c  (x 0   )   (x 0   )  c
2
2
2


  x 0   x 0  c    f (x )  
0
2
2
2
(21)
Price($/MW)
λc
bIL+2aILPILMax
K(2a(L-PIL)+b)
bIL
kb
b
L-PILMax
L
Power(MW)
Fig.2. Part of IL is interrupting
Since the differences between the samples and mean value is  on average,  will be added to the
2
value of f (x) in x 0 . According to these reasons, if the load is considered as a random variable in such
system, we can obtain Genco’s expected profit just by putting the mean value of load in Genco’s profit
function and adding (2ka  a) to it, and therefore no stochastic sampling is needed. Note that, the
2
expected profit of Genco increases by increasing the standard deviation of load. On the other hand,
Disco’s profit function could be obtained as follows:
2
Rd  c L  k(2aL  b)L  k2aL  L( c  kb)
(22)
2
Since Disco’s profit function is quadratic and L ’s coefficient is negative, Disco’s expected profit is
decreases by increasing standard deviation of load.
State 2-Part of IL is interrupting
In this state part of IL is interrupting with respect to the Genco’s bid (Fig.2), the condition for being in
this state is:
k(2aL  b)  b IL & k(2a(L  PILMax )  b)  2a ILPILMax  b IL
(23)
The amount of interrupted load is:
2a IL PIL  b IL  k(2a(L  PIL )  b)  PIL 
k2a
2a IL  k2a
L
kb  b IL
2a IL  k2a
L  
(24)
And therefore Genco’s profit will be:
2
Rg  k(2a(L  PIL )  b)(L  PIL )  (a(L  PIL )  b(L  PIL )  c)  ...
2
2
(25)
2
...  (1   ) (2ka  a) L  (1   ) (2  (2ka  a)  (kb  b)) L  ((2ka  a)   (kb  b)   c)
In this equation, 1   
2 a IL
2 a IL  2 k a
 1 . In this state profit is also a quadratic function of load.
Therefore, the relation between expected profit and load is like that in state 1, and since the coefficient of
2
L is smaller in this state, by increasing the standard deviation of load, the expected profit increases but
less than that in state 1. The profit of Disco in this state is:
Rd  c L  k(2a(L  PIL )  b)(L  PIL )  ...
2 2
2
...   k2a (1   ) L  (4ka(1   )   kb(1   )  c ) L  (kb  2ka )
(26)
2
Disco’s profit is also a quadratic function of load, and since the absolute value of coefficient of L is
decreased, by increasing the standard deviation of load the expected profit of Disco decreases but less
than that in state 1.
State 3- IL is fully interrupted
The condition for being in this state is:
k (2a(L  PILMax )  b)  2a ILPILMax  b IL
(27)
This state is like state 1, and the relation between expected profit and load is like that in state 1 and is not
obtained here for brevity.
State4- The relation with multiple suppliers
If there are multiple suppliers in the system, and each supplier submits following equation as its supply
function:
bid i  x i pi  yi
, i  1,..., n
(28)
ISO clears market by establishing Lagrange equation:
n 1
n
C   ( x i pi2  yi pi )   (L   Pi )
i 1 2
i 1
(29)
By using Lagrange multiplier method and some manipulation shown in appendix A, the  is derived:
n y
 i
i 1 x
L
i
 n

n 1
1
( ) ( )
i 1 x
i 1 x
i
i
(30)
This equation represents system’s equivalent supply function, which shows the relation between load and
market clearing price. With regard to such equivalent supply function, the profit of a supplier, for
example Genco1, would derive:
Rg1   p1  (a1p12  b1p1  c1 )
  x1 p1  y1  p1 
(31)
  y1
(32)
x1
Now we want to derive the relation between profit and load. Since profit is a quadratic function of load,
according to state 1, the coefficient of L2 will finally affect the expected profit of Genco. Now by putting
equations (30) and (32) into (31), the coefficient of L2 would finally be:
a1L2
L2
1

 L2 (
)(x1  a1 )
n 1 2
n 1 2 2
n 1 2 2
(  ) x1 (  ) x1
(  ) x1
i 1 x
i 1 x
i 1 x
i
i
i
The (x1  a1 ) is coefficient of L2 in state1, and (
(33)
1
) is added in the presence of other suppliers;
1 2 2
(  ) x1
i 1 x
i
n
therefore if the standard deviation of load is  , according to (21), the expected profit of jth Genco could
be calculated just by putting the mean value of load in Genco’s profit function and adding
(
1
1 2 2
( ) xj
i 1 x
i
n
2
)(x j  a j ) to it.
According to above equations we showed that in an unconstrained network, participants’ profit is a
quadratic function of load, and if the load uncertainty is modeled by a normal probability distribution
function, the expected profit of a Genco increases by increasing standard deviation of load and the reverse
occurs to Disco’s expected profit.
VII.
Problem solving algorithm
In previous sections participants’ profit model obtained and it was said that each participant for
determining its best strategy should solve equations (15)-(18). On the other hand, as mentioned before
participants could not predict the amount of load precisely, and they use a normal probability distribution
for modeling load uncertainty. Therefore, in modeling the problem, the amount of Pd,0 i in (13) is uncertain
and only its probability distribution is available. Since the problem is complicated, the analytical methods
could not solve it. Therefore, stochastic methods like Genetic Algorithm could be helpful. In this paper
fuzzy satisfying method and Genetic Algorithm are used simultaneously to solve bilevel optimization
problem and Mont-Carlo method is also used to model uncertainty in load. Since there are several
objectives we want to optimize, fuzzy satisfying method is applicable. In fuzzy set theory, each object x
in a fuzzy set X is given a membership function denoted by  (x) , which is corresponding to the
characteristic function of the crisp set whose values range between zero and one. In fuzzy sets the closer
the value  (x) to 1, the more x belongs to X. The x in this problem is participants’ profit. For every
participant a minimum and maximum profit should be obtained. Every participant’s maximum and
minimum profit could be defined either according to participant’s type, its ability to influence other
participants’ profit, its location in network, participants’ strategy, market regulation and etc, or by
obtaining each objective’s individual optimum value. After obtaining each participant’s maximum and
minimum profit, the membership function of each objective could be defined as follows:
0
 max
 Ri  Ri
 (R i )   max
min
Ri  Ri
1

min
Ri  Ri
min
Ri
max
 Ri  Ri
(34)
max
Ri  Ri
Note that maximizing all participants’ profits in (15) does not mean that all participants will finally profit
well, but rather if a participant has not the ability to influence market clearance point directly or
indirectly, other participants should not regard profit of such participant in their optimization problem, or
they can put maximum profit of such participant in (34) so low that it does not impose constraint to their
optimization.
Before calculating objective functions that are Genco’s and Disco’s profit, and then finding their
membership values, we must obtain Pg,i s, PIL,i s and LMPs. These parameters are obtained in lower level
sub-problem. On the other hand, in market clearing model, according to (9)-(14), K g,i and Pdg,i are the
inputs. Therefore, for every combination of K g,i s and Pdg,i s we will have a set of profits and by putting
these profits in their membership function we will finally have a fuzzy set of profits. In other words, for
every combination of inputs we will have a set of fuzzy profits in output. For comparing two fuzzy set of
profits, in order to find out which one is more desirable, the method used is to compare the minimum
members of fuzzy sets, and the set having higher minimum is more desirable, in other words the most
attention is given to the minimum value of each set. It can be formulated as follows:
N
Max imize [ min (  (R i ))     (R i ) ]
i 1,...,N
i 1
(35)
N
To circumvent the necessity to perform the Pareto optimality test, the term    (R i ) is added, where ρ
i 1
is a sufficiently small positive number [20]. For finding participants’ best strategy to maximize (35)
Genetic algorithm is applied. Each chromosome in this method is:
Chromosome   Kg1 ,...,Kg n , Pdg1,...,Pdg m 
The fitness function in Genetic Algorithm is (35), that determines which chromosome is more desirable.
For modeling load uncertainty, Mont-Carlo method is combined with above method. Accordingly, a
normal probability distribution is considered for each load. For each chromosome that enters the OPF,
sufficient samples are taken from these loads and the profits of companies will be calculated with respect
to sampled loads, and the average of these profits will be calculated to determine chromosome’s fitness.
Executing the algorithm involves the following steps:
1.
2.
3.
For every objective establish a membership function according to objective’s maximum and minimum value
like (34).
Specify a normal probability distribution for each load of system.
Create a Genetic algorithm whose chromosomes consist of Ng  Ndg genes, and each gene represents either
k g,i or Pdg ,i .
4.
Initialize the GA population and specify the number of generation allowed (the number of chromosomes in
each generation is denoted by Popsize)
5.
Set GA generation counter t gen  1
6.
Set the population counter t pop  1
7.
To calculate the fitness of each chromosome , do the following for chromosome t pop :
a)
Specify the number of Mont-Carlo trials denoted by TMont
b) Set MC trial counter t mont  1
c) Take sample of each random load
d) Solve market clearing problem (9)-(14)
e)
Save Pg ,i s , PIL,i s and LMPs derived from (d), and calculate Gencos’ and Discos’ profit with them
f)
t mont  t mont  1 . If t mont  TMont go to 7.c
g) Calculate the average profit of each t pop :
R
h)
t pop

1
TMont
(
T
Mont

R j)
j1
t pop  t pop  1 . If t pop  Popsize go to 7.
(36)
8.
Put expected profits derived in 7.g into (34) to derive fuzzy value of each expected profit.
9.
By putting fuzzy values derived in 8 into (35) a number will be derived for every chromosome t pop that
determines chromosome’s fitness.
10. t gen  t gen  1 . If t gen  Tgen go to11, else go to 12.
11. Perform standard GA operators such as crossover, mutation, etc and then go to 6.
12. Save the best chromosome derived in 9 as participants’ best strategies.
The flowchart of this algorithm is shown in appendix B.
It must be noted here that participants are considered risk neutral which means expected profit, or in fact,
their average profit in large trials is their deciding factor. Equation (36) could be changed depending on
participant’s risk sensitivity.
Table1
Generators parameters
ag
bg
($/MW2h)
($/MWh)
Gen1
0.1
Gen2
Gen3
Pmax
bid function
Coef. of L2 in (33)
30
140
0.2P+60
0.0147
0.09
21.5
160
0.23P+55
0.0156
0.12
22.2
150
0.27p+50
0.0121
VIII.
Numerical results
Part A:
In order to analyze what derived in section VI, we consider a system with 3 suppliers whose parameters
are shown in Table.1. By considering load as a random variable and using Mont-Carlo method, the
expected profit of Gencos are derived which are shown in Table.2, for example the first row of this table
shows that if the load is 300MW how much would be the expected profit of Gencos, and the last row
shows the expected profit of Gencos when the load is a random variable and its mean value and standard
deviation are 300MW and 45MW respectively. It should be noted that the expected profit of Gencos
increase with increasing in standard deviation of load. The expected profit of Gencos could also be
Table 2
Generators Profits with respect to increasing in standard deviation of load
Profit of Gen1($)
Profit of Gen2($)
Profit of Gen3($)
N(300,0)
3626.4
4879.6
4601.9
N(300,9)
3628
4881.5
4603.4
N(300,18)
3632.9
4886.6
4607.4
N(300,27)
3635.3
4889
4609.1
N(300,36)
3648.6
4903.4
4620.5
N(300,45)
3655
4910.4
4625.7
calculated by using what derived in (21) and (33). For example when the standard deviation of load is
36MW, the expected profit of Gen1 could be calculated according to (21):
R Gen1  3626.4  0.0147(362 )  3645.5
Which is approximately equal to the expected profit of Gen1 in Table.2. Therefore, the expected profit
could be calculated just by using above method, and therefore no stochastic sampling is needed.
Part B:
To examine the proposed algorithm a system shown in Fig.3 is applied. The information relating to
participants and transmission lines are shown in Tables (C-1)-(C-4) in appendix C. The following
information is assumed in this study:
 The simulation is running from Genco3 viewpoint
 Price that Discos charge their end customers for energy is c  95 $ / MWh

PILMax is one fifteenth of peak load
 Since with available methods the error in load forecast could be lessened to 3% [18], the standard
deviation of random load is assumed 3.5% in this study.
 The maximum and minimum profit of participants are assumed according to Table.3
Disco1
Disco2
Disco3
Genco1
Genco2
By using the method described in previous section the following results are obtained.
Genco3
Min Profit($)
-2500
-3000
-1000
-1500
-1500
-1500
Max Profit($)
4000
4500
2000
7500
7000
9500
Table 3
Maximum and minimum profits
Fig.3. Sample system
The best strategy for participants is according to Table.4, with respect to these strategies and load
uncertainty the probability distribution of Genco3’s profit is shown in Fig.4. Since participants are risk
neutral and their expected profit is important to them, the expected profit of participants must be
considered. The expected profit of participants with respect to these strategies is shown in Table.5, and
also the average amount of LMPs, lines flow, interrupted loads and power generated by generators are
shown in Tables (6)-(8) respectively.
Now if the amount of standard deviation of load is more or less than what the problem was solved with,
how would it affect profits. In other words, if participants have not paid enough attention in load
estimating and the standard deviation of load is a rough number, how much the error could affect
Table 4
Participants’ best strategies
KG2
KG4
KG5
KG6
KG7
KG8
PDG1(MW)
PDG2(MW)
PDG3(MW)
1.905
2.656
3.434
3.711
2.1
4.01
8.07
5.015
4.033
Table 5
Participants’ expected profits
Profit ($)
Disco1
Disco2
Disco3
Genco1
Genco2
Genco3
207.2
129.2
239.8
2212
2309
3031
participants’ profits.
Bus1
($/MWh)
Bus2
($/MWh)
Bus3
($/MWh)
Bus4
($/MWh)
Bus5
($/MWh)
Bus6
($/MWh)
Bus7
($/MWh)
Bus8
($/MWh)
101.7
100.12
99.2
97.32
95.17
94.96
98.08
95.34
Fig.4. Probability distribution
Table 6
of Genco3’s Profit
Average amount of LMPs
Table 7
Average amount of Lines flow
Line1(MW)
Line2(MW)
Line3(MW)
Line4(MW)
Line5(MW)
Line6(MW)
Line7(MW)
Line8(MW)
Line9(MW)
Line10(MW)
Line11(MW)
-15.57
9.48
-1.12
-15.25
8.74
-7.36
-8.94
7.29
-0.86
2.585
14.17
PG2 (MW)
PG4 (MW)
PG5 (MW)
PAverage
PG7 (MW)
PG8 (MW) andPIL1
(MW)
PIL2
(MW)
G6 (MW) amount
of generation
interrupted
loads
PIL3 (MW)
PIL4 (MW)
PIL5 (MW)
38.81
5.86
29.14
11.69
5.25
5.25
5.25
Table 8
24
15.37
5.25
4.05
In section VΙ the relation between participants’ profit and load was obtained, and it is said that with a fix
bid and uncertain load, the expected profit of a Genco increases by increasing in standard deviation of
load and the reverse occurs to a Disco’s expected profit. To analyze this influence in this system, we
increase the standard deviation of loads, and by considering that participants’ strategies are according to
Table.4, we will obtain expected profit of participants. The results are shown in Fig.5.
Fig.5. Gencos’ and Discos’ expected profit with respect to various amounts of standard deviation of load
As shown in Fig.5, the expected profit of Disco1 and Disco2 are decreased, Genco2’s expected profit is
increased, and the expected profit of Disco3, Genco1 and Genco3 are approximately stayed constant by
increasing the standard deviation of load. This trend in profits is inconsistent with what concluded in
section VΙ. This difference originates from the fact that what derived in section VΙ was in an ideal
network, but in this network we have line constraints that makes the relation between standard deviation
of load and expected profits to be dependent on participants’ location in the network. In above network
the line11 has imposed such constraint, and by congesting in most load samples taken has made different
LMPs in system. In the left side of this line, which the buses 5, 6 and 8 are there, LMPs are low, and in
the right side of this line, which the buses 1, 2, 3, 4 and 7 are there, LMPs are high, and the reason is that
supplying an additional megawatt of load at left nodes of the line decreases the congestion of line11 and
the reverse occurs in the right side, and accordingly such behavior in profits can be justified. Disco1’s
loads are located at buses 1 and 2; therefore, when the standard deviation goes up, it causes that there be
some high amount of loads among sampled loads that makes LMPs in these two buses increase
significantly, and hence Disco1’s expected profit decreases. Disco2’s loads are located at nodes 3 and 4,
and according to same reasons its profit decreases by increasing standard deviation. Since Disco3’s load
is located at node3, and supplying an additional megawatt of load in this node does not influence flow of
line11 considerably, Disco3’s expected profit has remained constant. Generators of Genco1 are at nodes 6
and 7. The expected profit of Genco1 is remained constant, because LMP at node 6 does not increase
considerably with increasing in load, and the generator at bus7 produces its maximum output for
approximately all sampled loads, and notwithstanding increasing LMP at node 7, the generator does not
have any more capacity to sell power. Similarly, Genco3’s expected profit does not change because its
generators are located at buses 5 and 8. Since there is LMP increasing in buses 2 and 4, and Genco2’s
generators are located at these buses, its expected profit increases.
According to above, load uncertainty could be classified into two types, if there is no constraint in the
system, section VI demonstrated that the profits of companies how would change, in such system since
the effect of load uncertainty on expected profits was determinable, participants can have a good
estimation of their expected profits without using any stochastic algorithm, and if they want to use
section VII’s algorithm in order to find their best strategy they can omit Mont-Carlo part in that
algorithm. If there is line capacity constraint in the system, the relation between standard deviation of
load and profits is highly dependent on participants’ location in the network. For instance, in the above
example the expected profit of Disco3 was not affected by Standard deviation of load, but Disco1’s
expected profit was sensitive to standard deviation of load due to its location in the network. Therefore,
according to such bidding, Disco3 could rely more on its expected profit, but Disco1could not be
confident enough and if he miscalculates the amount of load, its profit will change because of its location
in the network, and finally the proposed algorithm for bidding is better to be used in such system.
Conclusion:
Since in an imperfect competition each participant’s action has influence on market clearing price,
participants should consider other participants’ actions when maximizing their profits. In this paper a
method for participating in such competition in a pool market was proposed. In this method, because
power transaction in a power market occurs frequently, a strategy for playing cooperatively in the market
was suggested. Since one uncertainty in power systems is uncertainty of load forecasting, it can affect
participants’ bidding strategy. In this paper a method for bidding with consideration of load uncertainty
was proposed. The relation between participants’ profit and load in an unconstrained network was also
derived and it was shown that when the standard deviation of an uncertain load increases, the expected
profit of Gencos and Discos increase and decrease respectively, and it was also derived that the effect of
load uncertainty in the expected profit of a risk neutral participant is determinable. It was also shown
that, if there is line capacity constraint in the network, the relation between participants’ expected profit
and standard deviation of uncertain load has a different pattern, which differs with respect to
participants’ location in the network. Therefore, in such system participants should pay more attention in
effect of load in bidding strategy, and using the proposed algorithm could be helpful.
Appendix A:
ISO clears market by establishing equation (29) and by using Lagrange multiplier according to following:
  yi
c
 0  xi Pi  yi    0  Pi 
Pi
xi
(A-1)
c
0  L

(A-2)
n
n
P  0  L  P
i
i
i 1
i 1
By putting (A-1) into (A-2) we will have:
L
n

i 1
Pi  L 
n
  yi
i 1
xi

 L
n

n
x 
i 1
i

i 1
n
yi
1
 L 

xi
i 1 x i

n

i 1
n y
 i
i 1 x
yi
L
 n
 n i
1
1
xi


x
x
i 1 i
i 1 i
(A-3)
Appendix B:
Appendix C:
Specify Tgen , Tmont and popsize, and generate the first
population
tgen=1
tPop=1
tmont=1
Table C-1
Gencos’ parameters
Genco
No.
Gen
No.
Pmin
PMax
ag
($/MW2h)
bg
($/MWh)
cg
($)
2
G2
0
40
0.08
45.62
17.64
2
G4
0
58
0.11
35.35
31.6
3
G5
0
40
0.09
22.47
49.75
1
G6
0
50
0.095
23.37
89.62
1
3
G7
G8
0
0
24
60
0.085
0.078
33.47
21.39
24.05
79.78
Take samples from each random load
Table C-2
Discos’ parameters
Solve market clearing problem, obtain Pg,i , Pdg,I ,
PIL,I and LMPs, and calculate Gencos’ and Discos’
profit with them
Yes
tmont=tmont+1
tmont<Tmont
Disco
No.
Load
No.
Bus
No.
Load(MW)
aIL($/MW2h)
bIL($/MWh)
1
L1
1
N(35,1.23)
1
50.25
1
L2
2
N(27,0.95)
1
52.26
2
L3
3
N(35,1.23)
1
54.27
2
L4
4
N(35,1.23)
1
55.61
3
L5
5
N(35,1.23)
1
63.65
No
* N(35,1,23) = normal distribution with mean value 35
and standard deviation 1,23
Calculate the average profit for tpop by averaging
profits obtained in every tmont
Yes
tpop<Popsize
tpop=tpop+1
No
Table C-3
DGs’ parameters
Disco
No.
DG
NO.
Bus
No.
Pmin
PMax
adg($/MW2h)
bdg($/MWh)
1
DG2
2
0
10
0.09
34
2
3
DG3
DG5
3
5
0
0
9
8
0.09
0.09
34
34
Table C-4
Lines’ parameters
By putting profits of every tpop in (34) and (35) find the
fitness of that tpop
Perform standard GA operators such as crossover,
mutation, etc
Yes
tgen=tgen+1
tgen<Tgen
No
Save the best chromosome as participants’ best
strategy
Fig.B-1. Flowchart of proposed method
Line No.
From Bus
To Bus
X(p.u)
Limits
(MW)
1
1
2
0.011
30
2
2
3
0.018
30
3
2
4
0.03
20
4
3
7
0.022
40
5
7
4
0.015
30
6
4
6
0.03
30
7
4
8
0.03
40
8
8
6
0.0065
40
9
8
5
0.02
20
10
5
6
0.025
38
11
6
1
0.03
14.2
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